(1) Field of Invention
The invention relates to digital signal processing and, more particularly, to a method for estimating the noise power in some types of data that contains both signal and noise.
(2) Description of the Prior Art
The estimation of noise in the presence of a data signal is a large problem in many forms of communications and signal processing. The use of noise estimation in measurements taken from an array of receiving sensors can considerably enhance signal estimation. Indeed, estimation and statistical inference of noise are central to many disciplines that entail a series of observations, from which it is necessary to draw conclusions, estimate parameters and make forecasts.
Several conventional methods exist to suppress noise including the following:
1) The use of harmonic models.
2) Noise subspace characterization.
3) (FIR) filter modeling.
4) Data averaging.
5) Inter-epoch averaging, and.
6) Wiener filtering.
Nonlinear Time Series. Nonparametric and Parametric Methods, by J Fan and Q Yao, Springer (2003) offers a good overview of these conventional methods to handle nonlinear time series. Unfortunately, all the foregoing techniques suffer from at least one of the following limitations: the need for extended data acquisition, assumption of limited epoch-to-epoch variability, dependence on assumptions about the signal characteristics, and the inability to use conventional statistical approaches.
Data averaging is perhaps the most practical and widely used approach for estimating and reducing the noise with respect to the signal. For example, United States Patent Application 2004/0097802 by Cohen filed May 20, 2004 shows a method and apparatus for reducing contamination of an electrical signal. The electrical signal is digitized and averaged to obtain an estimated contaminating signal that is subtracted from the digitized electrical signal. This method is shown in the context of electrophysiological signals, such as EEG, ECG, EMG and galvanic skin response, and for elimination of noise associated with methods such as MRI.
To summarize the general approach to noise estimation by averaging, we start with the assumption that we have a data record comprising a group of data samples. A data record is usually divided into N equal intervals. If the data takes the form of a time series, each data sample x(ti) at time intervals consists of a signal s(ti) that is assumed to be correlated over the total duration of all intervals, and a noise component n(ti) that is assumed to be uncorrelated between any two intervals. Thus, at time ti x(ti)=s(ti)+n(ti).  (1)
There are two important statistics involved in the investigation of signal noise. The first is the mean, average, or expected value of a variable. This quantity is often mathematically denoted E(x), where x is a sample of the noise in question and E is called the expectation (average value) of the quantity inside the parentheses. This parameter is usually the signal that is being measured, to which noise is being added. The second statistic is the standard deviation of the noise. This is computed by subtracting the square of the mean from E(x2) and taking the square root. The standard deviation is a measure of the magnitude of the noise, whatever signal is present. If we have a sequence of N samples, the mean of any sample is denoted μ, where, by definition
                              μ          =                                    E              ⁡                              (                x                )                                      =                                          1                N                            ⁢                                                ∑                                      i                    =                    1                                    N                                ⁢                                  x                  ⁡                                      (                                          t                      i                                        )                                                                                      ,                            (        2        )            
Let the standard deviations for the signal and noise be denoted by S and σ, respectively, as follows (when both the signal and noise have zero means):
                              E          ⁡                      (                          s              2                        )                          =                                            1              N                        ⁢                                          ∑                                  i                  =                  1                                n                            ⁢                                                s                  2                                ⁡                                  (                                      t                    i                                    )                                                              =                      S            2                                              (        3        )            
and
                              E          ⁡                      (                          n              2                        )                          =                                            1              N                        ⁢                                          ∑                                  i                  =                  1                                N                            ⁢                                                n                  2                                ⁡                                  (                                      t                    i                                    )                                                              =                      σ            2                                              (        4        )            
If the signal is correlated between any two samples, and the noise is uncorrelated between any two samples (and the signal and noise have zero means), then
                                                                                          1                  N                                ⁢                                                      (                                                                  ∑                                                  i                          =                          1                                                N                                            ⁢                                              (                                                                              s                            ⁡                                                          (                                                              t                                i                                                            )                                                                                +                                                      n                            ⁡                                                          (                                                              t                                i                                                            )                                                                                                      )                                                              )                                    2                                            =                            ⁢                                                                    1                    N                                    ⁢                                                            ∑                                              i                        =                        1                                            N                                        ⁢                                                                  ∑                                                  j                          =                          1                                                N                                            ⁢                                                                        s                          ⁡                                                      (                                                          t                              i                                                        )                                                                          ⁢                                                  s                          ⁡                                                      (                                                          t                              j                                                        )                                                                                                                                              +                                                      1                    N                                    ⁢                                                            ∑                                              i                        =                        1                                            N                                        ⁢                                                                  ∑                                                  j                          =                          1                                                N                                            ⁢                                                                        n                          ⁡                                                      (                                                          t                              i                                                        )                                                                          ⁢                                                  n                          ⁡                                                      (                                                          t                              j                                                        )                                                                                                                                                                                                                      =                            ⁢                                                NS                  2                                +                                                      σ                    2                                    .                                                                                        (        5        )            
The signal-to-noise ratio is thus improved over that in each individual data sample by a factor N. It should be noted that a simple average as described above is not the only mechanism for “averaging out” noise. Other examples are weighted averages, moving averages, and moving-weighted averages. Underlying the application of averaging is a trade-off between the degree of certainty achieved and the number of samples that must be taken (and the time it takes to obtain them). It would be greatly advantageous to provide a more practical and efficient estimate of uncorrelated noise in data that achieves a high degree of certainty in a minimum number of samples.